Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $k = \dfrac{3}{3p + 7} \div \dfrac{3p}{7p(3p + 7)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $k = \dfrac{3}{3p + 7} \times \dfrac{7p(3p + 7)}{3p} $ When multiplying fractions, we multiply the numerators and the denominators. $k = \dfrac{ 3 \times 7p(3p + 7) } { (3p + 7) \times 3p } $ $ k = \dfrac {3 \times 7p(3p + 7)} {3p (3p + 7)} $ $ k = \dfrac{21p(3p + 7)}{3p(3p + 7)} $ We can cancel the $3p + 7$ so long as $3p + 7 \neq 0$ Therefore $p \neq -\dfrac{7}{3}$ $k = \dfrac{21p \cancel{(3p + 7})}{3p \cancel{(3p + 7)}} = \dfrac{21p}{3p} = 7 $